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April 29, 2008

Returning to Lautman

Posted by David Corfield

I mentioned in an earlier post that Albert Lautman had a considerable influence on my decision to turn to philosophy. I recently found out that his writings have been gathered together and republished as Les mathématiques, les idées et le réel physique, Vrin, 2006, a copy of which arrived through the post the other day. It’s remarkable how much contemporary mathematics Lautman covers – class field theory, algebraic topology, analytic number theory, etc.

At the same time as I was reading Lautman I became excited by category theory, via Colin McLarty’s Uses and Abuses of the History of Topos Theory, British Journal for the Philosophy of Science 1990 41(3):351-375, and Saunders Mac Lane’s Mathematics: Function and Form, and noticed an affinity with Lautman’s thinking, supported by a remark made by Jean Dieudonné in his 1977 Preface:

That Lautman worked with Claude Chevalley and Charles Ehresmann may not be unconnected.

The 2006 edition includes an interesting introduction – Lautman et la dialectique créatrice des mathématiques – by Fernando Zalamea, who devotes a section to interpreting Lautman through category theoretic spectacles. In successive posts I’ll take a look at how amenable are his mathematical examples to this treatment.

I’ll end now with a quotation Zalamea has selected from Lautman, which nicely expresses what might be done instead of Anglophone philosophy of mathematics:

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Re: Returning to Lautman

For Isaac Newton, mathematics was the response to the question: “How to understand the mind of God, and the divine laws which regulate the universe?”
(For the devout believer Newton, “divine laws” was meant literally, not metaphorically.)

A century later, mathematics at Oxford and Cambridge was a response to the question: “How to appreciate and approach the divine by reasoning in a manner supported by the divine?” Georg Cantor may have had similar motivations, as perhaps did Russian mathematicians in the first half of the 20th century, influenced as they were by a Russian Orthodox spiritual movement.

But the interesting thing about Lautman was that he didn’t remain at this ultimate question of what all of mathematics is responding to, but delved into specific theories as exemplifying certain themes, such as global-local relations.